Well-posedness in Sobolev spaces of the full water wave problem in 2-D

. We consider the motion of the interface of 2-D irrotational, incompressible, inviscid water wave, with air above water and surface tension zero. We show that the interface is always not accelerating into the water region, normal to itself, as rapidly as the normal acceleration of gravity, as long as the interface is nonself-intersect. We therefore obtain the existence and uniqueness of solutions of the full water wave problem, locally in time, for any initial interface which is nonself-intersect, including the case that the interface is of multiple heights. Oblatum 1b-II-1996 & XI-1996